A sadly under-rated game from the Japanese 2-D fighting game genre is Real Bout Fatal Fury Special. One character from that game, Blue Mary, has a tragic backstory in which she's a police officer on the trail of the killer of her partner and boyfriend?/fiance? from before the games started. That story isn't well reflected in her mannerisms, though, except in this "music video" that SNK obviously spent a lot of time on, and included in the Neo*Geo CD version of the game. This is her in-game theme music, expanded and now with lyrics (sung in Japanese, of course). This really highlights the pathos of her character, in my opinion, much more than anything in game could ever have done.
It's still a little cheesy; very anime in feel, and heck, the hardware that supported this video was debuted in the late 80s fer cryin' out loud, but given those limitations, it's surprisingly well done.